Die Lehrerin sagt, niemand müsse nervös sein, weil die Übung machbar sei.

They are Konjunktiv I forms. German often uses Konjunktiv I for reported speech or indirect speech.

So this sentence is presenting what the teacher says in a slightly more formal, reported way:

• more direct: Niemand muss nervös sein, weil die Übung machbar ist.
• reported/indirect: Niemand müsse nervös sein, weil die Übung machbar sei.

In English, we usually do not change the verb in quite the same way, so this is a very common point of confusion for learners.

Not as much as in formal writing, news reports, or more careful written German.

In everyday speech, many Germans would simply use the normal indicative:

• Die Lehrerin sagt, niemand muss nervös sein, weil die Übung machbar ist.

Or they might use a dass clause:

• Die Lehrerin sagt, dass niemand nervös sein muss, weil die Übung machbar ist.

So the sentence you have is correct and natural, but it sounds a bit more formal or written.

Yes. A very common version would be:

Die Lehrerin sagt, dass niemand nervös sein müsse, weil die Übung machbar sei.

The important difference is word order:

• without dass: niemand müsse nervös sein
• with dass: dass niemand nervös sein müsse

After dass, the verb goes to the end of the clause.

Because weil introduces a subordinate clause, and in German subordinate clauses put the finite verb at the end.

So:

• main-clause style: Die Übung ist machbar.
• after weil: weil die Übung machbar ist
• in indirect speech: weil die Übung machbar sei

The weil clause keeps subordinate-clause word order, and sei is the finite verb, so it goes last.

Because nervös is a predicate adjective used with sein.

German says:

• nervös sein = to be nervous

Here, müssen is the finite verb, and sein stays as an infinitive:

• niemand müsse nervös sein

So the structure is basically:

• subject: niemand
• finite verb: müsse
• rest of predicate: nervös sein

This is normal German word order with müssen + infinitive.

Because niemand is grammatically singular.

Even though it refers to no people at all, German treats it like a singular pronoun:

• niemand ist
• niemand muss
• niemand müsse

So müsse is the correct singular form here.

It is nominative, because it is the subject of müsse.

You can see the basic structure as:

• niemand = who is not supposed to be nervous
• müsse = the verb
• nervös sein = predicate

So niemand is doing the job that no one / nobody does in English as the subject of the clause.

Machbar means doable, feasible, or manageable.

It comes from machen plus the suffix -bar, which often creates adjectives meaning can be ...-ed:

• machbar = can be done
• essbar = edible, can be eaten
• sichtbar = visible, can be seen

So die Übung machbar sei means the exercise is not impossible or overwhelming. It does not necessarily mean it is easy; it means it is possible to do.

German uses commas to separate clauses.

Here there are two clause boundaries:

• Die Lehrerin sagt, ...
• ..., weil die Übung machbar sei.

So the commas mark:

1. the start of the reported statement after sagt
2. the weil subordinate clause

German comma usage is generally stricter than English comma usage, so learners often need to pay special attention to it.

A direct version would normally be:

Die Lehrerin sagt: Niemand muss nervös sein, weil die Übung machbar ist.

Or, if you want the teacher’s exact words more clearly:

Die Lehrerin sagt: Niemand muss nervös sein, weil die Übung machbar ist.

Compared with your original sentence, the differences are:

• muss instead of müsse
• ist instead of sei

That is because direct speech normally uses the indicative, not Konjunktiv I.

Lehrerin is the feminine form of Lehrer.

German often forms a feminine job title by adding -in:

• der Lehrer = male teacher
• die Lehrerin = female teacher

So die Lehrerin tells you the teacher is female.

Yes, in many contexts the meaning would be very similar, but the grammar changes.

With weil, the verb goes to the end:

• ..., weil die Übung machbar sei

With denn, the clause keeps normal main-clause word order:

• ..., denn die Übung sei machbar
• in ordinary direct speech: ..., denn die Übung ist machbar

So weil and denn can both mean because, but they do not behave the same way grammatically.